Can synchronous generator generates reactive power?

Yes! And it is simple as this:
In power systems, the reactive power flows from the higher voltage to the lower voltage, imagine one generator, your house, and a transmission line.

If alternator is Over-excited, it will deliver reactive power with lagging current.

while in Under-excited, it absorb reactive power with leading current.

But, it always(under or over-excited) deliver real power.

A synchronous generator is capable of producing active as well as reactive power. When the rotor is over excited, a synchronous Generator delivers reactive power . In general a synchronous generator works at 0.9–0.97 lagging power factor.