Welcome from Hanover Virginia
Welcome to the Forum. You will not find a greater wealth of information on anything Smith & Wesson, than right here on this Forum. Just for grins, try entering Smith & Wesson Model (insert any model number you wish) in your I-net browser..up will come some adds that say for sale like GB or GI...but invariably there will be several entries that link to S&W Forum, which are actual posts from our members, many of whom are real, true, experts in various Models and their variations.
I'm not one of those real, true, experts, however in answer to your question most of us do not know the exact date of manufacture at the factory, but we use the ship date to the original destination, which can be verified by Dr. Jinks, the SWHF Historian, by an official Factory Letter of Authentication, for a fee.
With that being said, I bought a Model 31-1 like yours, as new in box condition from a man who was settling his mothers estate. This particular gun was different from others in my collection simply because of the backstory. His mother was widowed at an early age when his father passed. Friends and neighbors convinced her to purchase a handgun for protection for herself and her young son. The son lived his whole life not knowing about the existence of the gun until the estate was his to settle.
Bottom line? I do not know exactly when this gun was manufactured, and I do not know when it shipped from S&W to the original local dealer and I don't have to send for a Letter, because I have the original sales receipt where the dealer sold this gun (serial numbered) to "Mrs. XXXXXX" on October 11, 1975. So in my case, this gun is at least 45 years old tomorrow, and that's close enough for me. The gun was never more than a couple miles from my house.
You may find similar provenance on yours, and again...WELCOME.