  We present a sample of 8498 quasars with both SDSS $ugriz$ optical and UKIDSS
$YJHK$ near-IR photometric data. With this sample, we obtain the median
colour-z relations based on 7400 quasars with magnitude uncertainties less than
0.1mag in all bands. By analyzing the quasar colours, we propose an empirical
criterion in the $Y-K$ vs. $g-z$ colour-colour diagram to separate stars and
quasars with redshift $z<4$, and two other criteria for selecting high-z
quasars. Using the SDSS-UKIDSS colour-z relations, we estimate the photometric
redshifts of 8498 SDSS-UKIDSS quasars, and find that 85.0% of them are
consistent with the spectroscopic redshifts within $|\Delta z|<0.2$, which
leads to a significant increase of the photometric redshift accuracy than that
based on the SDSS colour-z relations only. We compare our colour selection
criterion with a small UKIDSS/EDR quasar/star sample and a sample of 4671
variable sources in the SDSS Stripe 82 region with both SDSS and UKIDSS data,
and find that they can be clearly divided into two classes (quasars and stars)
by our criterion in the $Y-K$ vs. $g-z$ plot. We select 3834 quasar candidates
from the variable sources with $g<20.5$ in Stripe 82, 826 of them being SDSS
quasars and the rest without SDSS spectroscopy. We demonstrate that even at the
same spectroscopy limit as SDSS, with our criterion we can at least partially
recover the missing quasars with $z\sim2.7$ in SDSS. The SDSS identified
quasars only take a small fraction (21.5%) of our quasar candidates selected
from the variable sources in Stripe 82, indicating that a deeper spectroscopy
is very promising in producing a larger sample of quasars than SDSS. The
implications of our results to the future Chinese LAMOST quasar survey are also
discussed.
