Yes, of course. What you said implies, even without the $p=mv$ formula, that the momentum for a single object depends on its mass and velocity. Because you have said that the momentum is conserved, it guarantees that the speed won't change (assuming we know that the mass is conserved as well). It's called Newton's first law and it follows just from the qualitative comment about the momentum and momentum conservation.
You may also use the "very limited" description of the conserved momentum to say things about collisions etc. When you shoot into a big ball, the ball will get moving. It's a qualitative conclusion but I can deduce it from your limited information simply because the initial momentum was nonzero, so the final one must also be nonzero.
I don't know what's the point of hiding $p=mv$ but even without this formula, the momentum conservation law is, much like any conservation law, nontrivial. According to Noether's theorem (which is probably not understood by any person from the set of those who don't know $p=mv$, but let me mention it, anyway), the existence of the momentum conservation law is inseparably equivalent to a symmetry of the laws of Nature, namely the translational invariance. And this is a very nontrivial insight, indeed.
In fact, Noether's theorem doesn't even give you any "explicit" formula such as $p=mv$ in the most general case: one needs to know some specific things about mechanics to know or derive this formula. So it's helpful to know at least one simplest example such as $p=mv$, but it is not true that the momentum must always be something like $p=mv$. The momentum density (the stress-energy tensor) of the electromagnetic field looks very differently than $p=mv$.