RAW? No, you cannot.

HellSaint's answer covers this well. If you declare a hostile action (like casting hold person), the game drops into rounds and turns:

1. Determine surprise.
2. Establish positions.
3. Roll initiative.
4. Take turns.
5. Begin the next round.

What can the Alert character even do?

If the Alert character goes first...

...describe the environment as you normally would.

DM: As you walk through the market, your stomach suddenly drops and the hairs on the back of your neck stand on end. You don't immediately see anything unusual, but you know this feeling - your life is in imminent danger. What do you do?

PC: I look around to see who or what is trying to kill me.

DM: (Hmm...this could be Perception vs Stealth, but the sorcerer isn't hiding. It's more about blending in.) Make an Insight check as you look around the crowded market. This will take your action.

PC: Sounds good. (rolls) I got a 21.

DM: (The sorcerer gets a 17 on Deception; not good enough.) The crowd ebbs and flows, but one figure stands out - a half-elf in a tattered robe is staring at you. You can see hatred burning in their eyes.

Where it goes from here is likely up to the PC's class, equipment, and personality. Here are some examples:

• Any PC could dive through a nearby window to try to break line-of-sight.
• A wizard could (more elegantly) misty step through said nearby window.
• A fighter could move through the crowd, Action Surge, then attack the sorcerer.
• A rogue could use Cunning Action to Hide behind a nearby market stall.
• A firbolg could use Hidden Step to turn invisible.
• A sorcerer could use Quickened Spell to get the jump on the NPC sorcerer.

Even if the PC failed the Insight check, taking cover or hiding are reasonable responses that may save the PC's life.

If the NPC sorcerer goes first...

...the Alert feat still matters:

If you're surprised, you can't move or take an action on your first turn of the combat, and you can't take a reaction until that turn ends.

Thanks to the Alert feat, the PC can still take a reaction.

• A caster could possibly counterspell. If the sorcerer's spell required no material components (like scorching ray), then counterspell definitely won't work against Subtle Spell. With material components or an arcane focus, it's a trickier call.
• A PC with a staff of the magi could use its Spell Absorption trait. Unlike counterspell, Spell Absorption does not require that you see the spell being cast. This also works with a rod of absorption.
• An Arcane Trickster rogue could use Spell Thief (which also does not require you to see the spell being cast) to attempt to disrupt the spell.

Conclusion

Your best bet is to use the initiative and surprise rules as-written. If the Alert character goes first and decides to crash through a nearby window, the NPC sorcerer has an interesting decision to make. Do I target one of the other characters, or melt back into the crowd to strike another day?

The transition from not-Combat to Combat happens when the DM says it does. It doesn't have to happen on "turn boundaries".

Turn boundaries are an abstraction to keep combat simple.

Someone with the alert feat cannot be surprised (well, except by a plot twist or the like, but they still get to roll initiative). And entering combat doesn't happen with no reason.

You are the DM, but you should respect the alert feat. There is nothing you are allowed to do that surprises an alert PC (in the sense that someone else acts while the PC doesn't get a chance to beat them to it).

At the same time, subtle spell means that the sorcerer is not doing anything detectable to cast the spell.

Which leaves the spell.

If the spell has no visible effects, it isn't detected. But hold person does.

Combat is about to begin because someone is being held. That is the first hostile act detected.

Start combat as that starts to happen, but before the save is resolved.

Tell the player "you feel your muscles start to tighten and lock up. You cannot see a cause." then roll initiative.

If the sorcerer wins, they must make a saving throw or be held. If the player wins, they get to act before they have to make a saving throw. If they break line of sight, gank the sorcerer, drink a potion of freedom of movement, or whatever -- they bypass the threat.

Combat has begun after the spell starts to land, and before the saving throw. This is normally a period of time that combat skips over.

If the player isn't a target, have them roll an insight check to see if the person who is targeted is starting to lock up. Failing that, they feel the situation is funny and dangerous, but not why.

Similarly, you can start combat with arrows in the air, a knife being stabbed, etc.

Rule 0: You can. But you can't - and you probably shouldn't.

Combat starts when the intent to attack starts (attack or do something else harmful). You should roll initiative, and, if the player character rolls higher than the sorcerer, they have the opportunity to do something. This is the whole point of the Alert feat and the initiative system: it doesn't matter who declares or intends to attack first, it matters who actually acts first, and who actually acts first is determined by initiative. See this Q&A for better understanding as well.

Important to mention: It doesn't mean they necessarily notice that the Sorcerer, specifically, is doing something¹. But they should notice that something is off, that something dangerous is happening. They should have the chance to, at least, run to somewhere safe where the sorcerer doesn't have line of sight, or anything similar. Both these answers tackle on how to describe that the PC is in combat without giving it away that the Sorcerer is shady from the beginning.

Alternatively, you can think in these terms: is the character (and the player) going to be "surprised" (in the usual, English sense) by being paralyzed without any warning? Yes. But they shouldn't be surprised, ever, so there you go.

"Aw but then I can't ever surprise them in any sense?" - Yeap. They spent an entire ASI for that. These don't come by often. They could be getting +2 in their primary AS, they could be getting Lucky or Great Weapon Master, but they chose to get Alert so they wouldn't get surprised. Alert is already not an amazing feat unless you are playing an Assassin Rogue. Please don't make it worse with your rulings.

As a final comment on DMing: even if your reading of the rules disagrees with mine, I would still urge you to not do that. As the DM, you have infinitely many NPCs with infinitely many options. It is not a huge loss for you to have your NPC unable to do what they intended/wanted to do. The player only has that character (most likely). I would certainly avoid anything that makes their character building choices seem meaningless or frustrating, and bypassing an entire feat they chose is, in my experience, one of these things that will probably make them feel frustrated and like they chose wrong. Instead, personally, I would use this as an opportunity to reward their building choice: let them have the chance to act before the sorcerer, transform an encounter that could start insanely difficult in an easier encounter because they chose that feat and let them feel rewarded for it.

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¹ To further clarify: it doesn't even mean the characters notice the Sorcerer, or that they are being targeted, or anything else. How much they notice is up to you, as the DM. It may range from "You sense something is off. Period." to "You notice a Sorcerer looking in your direction and suspiciously hiding something in their hand, they may be holding an arcane focus and casting a spell". But there is a floor of awareness which, in my opinion, is sensing something is off, no matter how vague it is.

Surprise is determined at the start of combat, so Alert should not matter

PHB 189 states (in the box text "Combat Step by Step"):

1. Determine Surprise. The DM determines whether anyone involved in the encounter is surprised.

Surprise (PHB 189) is simply the following:

If you’re surprised, you can’t move or take an action on your first turn of the combat, and you can’t take a Reaction until that turn ends. A member of a group can be surprised even if the other members aren’t.

Surprise is determined once combat is initiated.

So, when the Alert feat says "You can't be surprised while you are conscious (PHB 165)," what it's really saying is that you are immune to the above surprise mechanic.

In your scenario, the casting of the spell seems like it would initiate the potential combat, so the player with Alert should not be able to interfere with your casting.

Additionally, no one should be able to tell that the Sorcerer actually cast a spell unless there are clues leading to that conclusion, like:

• They know the Sorcerer are capable of using the Subtle Spell metamagic
• The Sorcerer and the target were in a heated altercation before the spell was cast
• The Sorcerer is the only one around that could have cast the spell

Further, if the party can already see the Sorcerer before the spell has been cast, then the Alert feat's immunity to surprise is useless since the surprise mechanic requires one party or another to be sneaking or otherwise be hidden. If both sides are visible to each other, no surprise would take effect.

The GM determines who might be surprised. If neither side tries to be stealthy, they automatically notice each other. Otherwise, the GM compares the Dexterity (Stealth) checks of anyone Hiding with the passive Wisdom (Perception) score of each creature on the opposing side. Any character or monster that doesn’t notice a threat is surprised at the start of the encounter.

"Can I do this without asking my players to roll for initiative?" Yes, but initiative should probably start immediately after depending on the circumstances as discussed a bit above.