Welcome to the forum. That is a very neat old S&W and since it belonged to a family member should be a valued heirloom. Should you decide to remove the stocks to check for a serial number, be very careful to use the correct size blade screwdriver when you remove the screw. Damaging the screw will decrease the value.
To remove the stocks, loosen the screw and while it is still engaged in the opposite panel, give the screw a gentle tap with the right stock panel facing down and over a soft cloth. This will generally loosen the right stock panel from the frame enough to enable you to remove it. The left panel, if not all ready loose, can then be tapped through the frame on its inside to remove it from the frame. Stocks that have been on a gun for a long time are sometimes stuck to the frame by years of oil and gunk and sometimes need gentle persuasion. Work slowly and gently as you do not want to chip the wood.
Once the right panel is removed, check to see if there are any numbers either stamped or written on its inside surface. Usually, this is the serial number of the gun that the stocks were originally matched to. If the numbers match the numbers on the butt of your gun, that is great. If not, then it falls into the category of many S&W's out there with non matching stocks. Others have stated that the stocks in the photo are from a later time frame than your revolver and I will defer to their observations.
While you have the stocks off, give the frame a good cleaning with Hoppes or something similar and then a good coat of Renaissance wax if you have any. If not, a little oil should keep the frame from rusting under the stocks.