Time in which individual attention is given to an otherwise neglected child or partner.
It will come as no surprise to read that this is an American phrase. It dates from the 1970s, the time when the notion that parents could 'have it all', that is, a successful career and a happy home life, was gaining popularity.
The first use of the phrase that I've found is from the Maryland newspaper The Capital, January 1973, in a piece headed 'How To Be Liberated':
The major goal of each of these role changes is to give a woman time to herself, Ms. Burton explained."A woman's right and responsibility is to be self fulfilling," she said. She gives "quality time" rather than "quantity time" to each task, whether it be writing, cleaning the house or tending the children.
See other phrases that were coined in the USA.