Expected inflation in the real interest rate equation

It is the expected inflation rate over the life of the instrument. So if it is a 10-year bond, it is expected inflation over the next 10 years.

If you use future values of the price index to determine “expected” inflation, you are assuming bond investors can predict the future perfectly. Based on my experience, that assumption is implausible. It could be justified, but one would really need to be careful about how the data are interpreted.

Note: I missed a clarifying comment. The original poster was only interested in whether to use the next quarter, or +4 quarters for “inflation expectations.”

For determining the real rate, it should still be matched to the term of the bond. E.g., subtracting 3 months inflation from a 10-year bond makes little sense.

For a model, 1 quarter ahead is used in some contexts (linearisations of models), otherwise, it is the entire model horizon. E.g., a model that projects a solution out to infinity, it’s the entire inflation trajectory.

Your formula isn't quite right. If $r$ is the nominal interest rate, and $pi$ is the inflation rate, the real interest rate is $frac {1+r}{1+pi}-1$. The formula $r-pi$ is approximation for small rates. (Note: for all of this, I'm using "rate" to mean the multiplicative factor minus one. So if the balance is multiplied by 1.2, the rate is 0.2.)

The inflation should be over the period over which the interest rate is being calculated. It's like to like: the real interest rate at a particular time is given by the nominal interest rate at that time compared to the inflation at that time.

The fact that you think you are wondering whether you should use the future inflation suggests that you may be confusing the spot inflation rate with the Consumer Price Index. You can get the real interest rate by taking the total nominal interest times the ratio between the CPI at the end of the period to the CPI at the beginning. That is, if $r$ is the total nominal interest, $CPI_i$ is the starting CPI, and $CPI_f$ is the final CPI, then the total real interest is $(1+r) frac {CPI_f}{CPI_i}-1$.