I think there are actually *three* possible things that you might be asking, but the answer to all of them is no.  Suppose that G is a strong generator in a cocomplete category C.  Then you can ask:

1. Is every object X of C the colimit of G over the canonical diagram of shape $(G\downarrow X)$?  (If so, then G is called *dense* in C.)

1. Is every object *some* colimit of a diagram all of whose vertices are G?  (If so, then G is called *colimit-dense* in C.)

1. Is C the smallest subcategory of itself containing G and closed under colimits?  (If so, then G is a *colimit-generator* of C.)

The category of compact Hausdorff spaces is a counterexample to the first two.  It is monadic over Set (the monad is the ultrafilter monad, aka Stone-Cech compactification of the discrete topology), and hence cocomplete, and the one-point space is a strong generator.  But any colimit of a diagram consisting entirely of 1-point spaces must be in the image of the free functor from Set (the one-point space being its own Stone-Cech compactification), and hence (since that functor is a left adjoint and preserves colimits) must be the free object on some set.  However, not every compact Hausdorff space is the Stone-Cech compactification of a discrete set.

As Todd pointed out in the comments, though, CptHaus is the colimit-closure of the one-point space, since every object is a coequalizer of maps between free ones (because the category is monadic over Set); thus it isn't a counterexample to the third question.  Counterexamples to the third question are actually much harder to come by, and in fact if you assume additionally that C has finite limits and is "extremally well-copowered", then it is true that any strong generator is a colimit-generator.  I think there's a proof of this somewhere in Kelly's book "Basic concepts of enriched category theory," and a brief version can be found [here](http://www.math.uchicago.edu/~shulman/exposition/generators/generators.pdf).  However, without these assumptions, one can cook up ugly and contrived counterexamples, such as example 4.3 in the paper "Total categories and solid functors" by Borger and Tholen.