Yes but both glue and screw the cement backer board down to the plastic laminate countertop, using either thin set or construction adhesive, before laying the tile. This will make the backer board more rigid, and prevent the tile from cracking or coming loose.

Gluing the backer board to the countertop is particularly important when using 1/4" thick cement backer board, since it's more flexible than 1/2" backer board.

The best thing to do would be to remove the counter top that you have now. The cultured marble that you have now was not designed to take the additional load of tile on it.