Is a global optimal solution of a convex problem always unique? I do not have a specific problem. Could a convex optimization problem (not strictly convex) have alternate solutions?

Yes. Take for instance $$\inf_x \max\{|x|-1,0\}$$
The minimum value is clearly zero, but 
$$\mathop{\textrm{argmin}}_x \max\{|x|-1,0\}=[1,-1]$$
Notice that the set of optimal points is an interval. In the general, multivariate case, the set of optimal points for a convex optimization model is always a convex set (including possibly the empty set or a singleton).
