No, strings do not replace the quark model. Quarks are part of the present day successful theory called the Standard Model. And string theory does not propose to replace that but to explain it. The basic idea is that the quarks, and other elementary particles, are actuallly vibration modes of tiny, very very tiny strings, or other things called branes.
To give you an idea of what a vibration is, a note, such as middle C, is a vibration mode of a string in a piano. That same string has other modes, overtones of C, and each one of those tones is a vibration mode of that string.
So the tiny strings (BTW, string theory doesn't say what they are made of) vibrate in different modes, which are moreover quantized so the modes don't vary smoothly among themselves but come at separate energies, these quantized modes in some way constitute the particles. The way they do that has not really been fully explained by the string theorists, but they insist they are making progress and will be able to explain it real soon now.
So bottom line the quarks would still be used, just as Maxwell's equations of electromagnetism are still used in spite of the development of quantum electrodynamics, but another level of understanding would have been added.