If by spectrometer you mean REFRACTOMETER, then by all means yes, many of us use them to check OG and gravity of runnings during mashing and sparging.
I also have learned to trust ONLY the BRIX side of my refractometer. Many of the cheaper models that can be found online are susceptible to some discrepancy between the correlation of one scale to the other.
As far as checking FG, alcohol causes some discrepancies in the readings which require a conversion that can be found on this site and many online references.
IMHO, just get yourself a hydrometer (if you do not already have one) and use it to check FG.
Another point to make about refractometers (that I used to tell my Chemistry students) is that it is an instrument and being so, it is only as accurate and smart as the operator.
There are specific "must do" things to get accurate readings from a refractometer, especially a $40 Chinese model.