Liara knew that her grandfather is the Master Chief and she also knew that she had to find him herself when she had seized the chance to do so without her team.

But, as to the question why the Master Chief was her grandfather, only her mother and Dr. Hasley knew the answer to it. But, eventually, Liara will learn more about it herself.

She was often sent on solo recon missions because as the UNSC knew, Spartans tended to be one man armies all on their own. And it's always true about the Spartans as well, being one man armies unto themselves of course.

She knew that many people had hoped that she'd find the Master Chief, or at least a trail, but she knew that she would find her grandfather in time. and after both Noble and Nova remained at base till they were sent on other missions because she has to become a true telepath among her mother's people.

And to do so, she has to go through the remembrence and to do that, she has to find the wisest of the fauns to perform the ceremony and help her become more attuned to it as well.