[Edited to remove false information]
In his paper Overrings of Prüfer Domains,$^1$ Gilmer constructs a ring which is a locally a DVR and has all but one maximal ideal finitely generated. It is the last example in that paper. In general, "most" almost Dedekind domains (aka locally DVR rings) which are non-Noetherian will have some sharp primes. The literature on almost Dedekind domains is a good place to look for examples of the sort in question.
However, there cannot be any non-noetherian 1-dimensional local domain with a finitely generated maximal ideal. This follows, as user 26857 noted in the comments, from Cohen's theorem which states that a ring is Noetherian iff its prime ideals are finitely generated.
I am not sure whether there can be f.g. maximal ideals in non-Noetherian $1$-dimensional semi-local domains.
$^1$ Overrings of Prufer domains, I, J. Algebra 4 (1966), 331-340