It kind of depends… sometimes the hardcover is released simultaneously with a paperback of the same size, which is sometimes called an “export edition”. More and more, these large paperbacks are replacing hardcovers entirely. They will have the same page count, because they literally use the same pages.
Some novels, usually literary novels, will be released in trade paperback (the trim size will be smaller than the hardcover, but larger than mass-market) some months after the hardcover. These will likely have a different page count.
Finally, the novel will be released in a mass-market, or pocket book, or rack size, format. This is the smallest format, and the page count will likely be different from both of the previous paper editions.
Classically, the release of these paperbacks is staggered, but it is increasingly common to have multiple formats released at the same time, or for one or more of these steps to be dropped. Authors who have had a long career, and therefore a trained audience, will probably be faithful to the old model, so look for authors like Danielle Steel or John Grisham for mass-market page counts. Otherwise, be sure to check the dimensions of the paperback to be sure that it’s the type of paperback that you want to compare to the hardcover.