Let $X$ be a compact connected manifold (with or without boundary) and let $H_1(X)$ denote its first Cech integral cohomology group or, equivalently, its first cohomotopy group. Is it true that $H_1(X)$ is finitely generated even if $X$ is not triangulable? Could you please also provide a reference?
Manifolds are ENRs and a compact ENR is a retract of a finite simiplicial complex. This is proved in corollaries A.8A.9 of Hatcher's algebraic topology text. Retracts of finite simplicial complexes have finitely generated cohomology (of any flavor). 

