In general, no and it will happen when there is something actually written to the page (which will cause a page fault and the kernel will have to somehow materialize the page). This works the same way regardless of how /proc/sys/vm/overcommit_memory is configured, the setting only affects how kernel tracks how much memory it is going to need in the future. (Obviously if we talk about malloc() this is a slight over-simplification as most malloc() implementations will write some kind of book-keeping structure and thus dirty some of the allocated pages)

Whether swap is available is more or less irrelevant for this behavior. The only thing that swap changes is that kernel is then able to “clean” dirty anonymous pages by writing them out to swap.