No, payday lenders will not contact your employer. So you can be rest assured that they will not call up your boss and say “Hi, I am calling from XYZ Loans to speak to …”
However, during the application stage, some lenders (not all) may call your workplace in order to confirm that you are currently employed there. The lender may ask to speak to you and that is all. If the number you provided matches, this is a good indication that you work at that company and are therefore receiving a regular income. This in turn, gives the lender confidence that you will be able to keep up repayments and repay your loan on time.
Lenders are aware that applying for payday loans is a sensitive issue and they will not state to anyone else where they are calling from.