This is true in greater generality. If $M$ and $N$ are locally compact, and the fibers of $f$ are compact connected and $f$ is a quotient map then f is proper. This falls under the rubric of monotone light factorization. Look at the book of G.T. Whyburn and E. Duda titled Dynamic Topology. Monotone Light factorization says if a map between locally compact Hausdorff has the property, that all connected components of all fibers are compact, then the map can be factored as a proper map onto a space with connected fibers and this space maps to the co-domain with totally disconnected fibers. Thanks to Dan Petersen and Richard Andrade for correction.
Addendum:
Let a connected component of the fiber of a map be called a level.Suppose X and Y are locally compact Hausdorff spaces and F a continuous map such that all levels are compact .Suppose the equivalence relation R on X is given by saying x and y are equivalent if they lie in the same level, then R is a proper equivalence relation
and the map from X to X/R is proper with connected fibers and F factors through X/R to Y as a map with totally disconnected fibers . In the case of the question of
the OP the equivalence relation R and the one given by f coincide and therefore the map is proper

typeof thing that has been discussed at math.stackexchange, e.g., here: math.stackexchange.com/questions/7096/… If OP cannot construct an argument from that or similar threads, he should probably ask over there. (I happen to think it's a good question.) $\endgroup$Pleaseif after some time you do not get an answer in math.stackexchange.com it can be sensible to reask here, butalwaysprovide links to the other question in both sides, andpleasedo not delete questions which have answers on which other users have spent time and energy. $\endgroup$1more comment