First morning urine is not essential unless you are diabetic and checking blood sugar or unless you are concerned that you might be pregnant. It is not necessary for a routine urinalysis or drug screen.
First Morning's urine is only needed for pregnancy tests, but even then they will still work just fine if it's not first morning urine. So the answer is no.
If your doctor has advised to bring it for tests, you do it so. The first morning urine is generally needed to determine hormonal levels and sugar levels.
