No they don't.
Around 2 million people take beta-blockers in the Uk , at a a guess and i have never heard of weight increasing following their use although never say never in medicine.
It is a common and dangerous habit to blame symptoms on a particular drug.
What is more worrying is that in the case of your cousin and if she is taking them for heart problems that her increase in weight is due to fluid retention indicating incipient congestive cardiac failure.
If this IS the case, then she should see her GP. rather than blame betatblockers.