Did America's brief "nuclear monopoly" have any effect on the USSR at all?

It motivated them to make their own bomb!

But if you're asking, "did it deter them from acting dangerously?"... it's very hard to say. To answer that is to pose a hypothetical: if the US didn't develop nuclear weapons, would the USSR have done anything differently? It's very hard to say.

We can say the following with some confidence:

The USSR had many reasons to be done with war at the end of WWII — there was a lot of rebuilding to do.
The USA did not have a significant nuclear arsenal during most of the period of its monopoly. It would have been very difficult to deploy against the USSR. It would not have been a "kill them all in one wave" sort of thing — it would have been World War II plus several nuclear detonations. The bomb didn't become an "36-hour war" sort of thing until the 1950s, when the US had more weapons, more means of deploying them (including overseas basing), and lots more options in terms of what to do with them (for the entire period of the American monopoly it essentially only still had the same kinds of bombs it used against Nagasaki and the same kinds of planes, the B-29).
The USSR still did some pretty aggressive things despite the US monopoly. It cheerfully made sure that any countries it occupied became puppet states, despite promises to the contrary. It did the Berlin blockade in 1948 — about as risky a maneuver as you can do short of actual war (and indeed, the US feared it might end up in a war). It worked hard to assist Communist forces in China and North Korea, against US interests. And so on.

Would the Soviets have gone further without the US monopoly? Maybe — maybe they would have thought it prudent to keep moving West. Personally I sort of doubt it, as it would involve some great risk to themselves. As it was, they were not deterred or cowed to an unlimited amount by the US monopoly — they kept things shy of actual war, but did a lot of maneuvering in their interests, even some quite dangerous things.