Since you are using the INT function, dividing by three, and subtracting one, the first three months of the year WILL return 0.
The CHOOSE function can work very well for this, but the trick is getting the list of values in the correct order, as the function will choose the value from the list in order based on the number of the month. January = 1, February = 2, March = 3, etc. So, it will choose the 1st value for January, the 2nd value for February, and so on. Since your fiscal year starts in April, January, February, and March are the fourth quarter; therefore, your first three values should be 4, followed by three 1s, three 2s, and three 3s, like this:
="Q" & CHOOSE(MONTH([Date of Inspection]),4,4,4,1,1,1,2,2,2,3,3,3)
January, February, and March will return Q4; April, May, and June will return Q1; July, August, and September will return Q2; and October, November, and December will return Q3.
I got around checking for a null value by making it a required column, and don't have to use an IF statement.