All over Spain you can find quiet areas and small beaches if you have the time to explore - i guess a smaller, quieter beach is what you are looking for as a couple, rather than one with too many families, crowds of people doing watersports etc. But one area I would recommend is the province of Girona, in the Catalan region.

Girona itself is a lovely town and not too far from Barcelona, but a big attraction of the area is the coast. If you go out of high season and head for small resorts like Begur or Tamariu, you will find small beaches with rocky coves and lovely walks along the cliffs. At the right time of day you may find a small cove with hardly anyone in it, but on the main beaches it's not too crowded either (at least, it wasn't in September, when I was there.) However, although the beaches are relatively quiet they often have refreshment kiosks, and the shops, restaurants etc are not far away.

Have a great holiday.