Designing on-chip antenna
Is it also called microchip antenna ?

no. May call it a microstrip antenna, but I'd recommend just calling it PCB antenna.

Can anyone provide me details and/or steps as to how to design on-chip antenna?

That is a complicated question, and you've literally defined none of the requirements, so I'll have to guess you don't really know how antennas work. So, start with a bit of basic antenna theory. Later you can design your own antenna. Designing takes understanding.

I want to design my own PCB having an on-chip antenna for GPS module. It is possible, right ?

Hm, probably not a good idea: GPS antennas are typically relatively selective, so you'd want a design that is proven to work. Also, if you're implementing it directly as traces on the PCB, it'll probably be rather large. You'll also need to match your own antenna to some (most likely, 50Ω) transmission line impedance, requiring more components, and fine-tuning.
Also, GPS antennas are typically active devices containing an amplifier.
So, really, buy an active GPS patch antenna. It solves all these problems at once, and due to the fact that it saves both board space and additional components, will not only perform better, but quite likely even be the cheapest you'll get.