Topological indices (TIs) have an important role in studying properties of molecules. A main problem in mathematical chemistry is finding extreme graphs with respect to a given TI. In this paper extremal graphs with respect to the modified first Zagreb connection index for trees in general and for trees with given number of pendants, for unicyclic graphs with or without a fixed girth and connected graphs are determined, using methods with higher degree of generality with respect to the transformation techniques usually used in such context. These graphs are relevant for chemical studies.