Well, the 737 airframe has been used by Indonesia for the MPA role since the 80s, so it's already been doing that role for about 30 years. It's not too impractical to think that it could have been started earlier. The 737 is--in some ways--an evolution of the 707 series that started with the Dash 80 back in 1954. The Dash-80 was designed as dual-use, spawning the KC-135 and 707 at basically the same time, so considered that way, the 737 is an evolution of an airframe designed from the start with the possibility of military applications, yes.
So, could the 737 become the maritime patrol plane of the US Navy in the P-3 period of the Cold War starting in the 1960s? To be honest, no, I don't think it could have. I think it could have been an MPA much sooner than it did, but it wouldn't have been nearly as capable. The 737 was initially designed for relatively fast, short-haul work at medium altitude. The was basically the opposite of what a Cold War MPA needed to be. An MPA in the 60s-90s needed to be able to fly low and slow for VERY long period of time. The 737 could do exactly none of these things well.
Since then the 737 has gotten to be much larger, able to carry a lot more fuel and equipment. It's really a beast of a plane compared to its 1960s forefather. Jet engines are now a lot more fuel efficient and reliable. In the 1960s, a twin-jet plane couldn't have cruised 1,000 miles out to sea for 6 hours. Even if it could, the very real possibility of engine failure meant that you wouldn't want to attempt it. That's no longer the case.
Sensors have gotten much better and submarines far less numerous since the 1990s, as well, placing a greater emphasis on surface patrol. For both of these reasons, it's better to conduct these patrols at medium- to high-altitude rather than low-altitude.
You couldn't have really had a 737-based Poseidon in the 1960s or even the 1970s. Even in the 1980s, such an aircraft would be inferior to the 20 year old P-3 then in service. You COULD have had a 707-based aircraft, but it would have been much closer to a Nimrod than a P-3, in all likelihood.
I hope that helps!
Cheers,
Logan