"quickly familiarize" is correct and usable in written English.
You would generally use it when talking about someone becoming familiar with something in a short period of time. For example: "She quickly familiarized herself with the new software program and was soon able to use it with ease."
One will quickly familiarize oneself with that button, because the designers have deigned a little 16-inch strip the standard helping of paper.
This technique is commonly integrated with web sources to quickly familiarize visitors with the content via key words.
The authors conclude that to successfully respond to shifting situations, evaluators must become effective inquirers, able to accurately and quickly familiarize themselves with the changing interests of stakeholders, who are often associated with emerging technologies.
The new hardware allowed for processes which the team had never used in PlayStation 2 game development and required them to quickly familiarize with the new techniques; for example, parallel processing and pixel shaders.
Gain familiarity with a category -- i.e. social networking -- and you gain the basic working knowledge needed to quickly familiarize yourself with any major new market entrants, or feature introductions, as they roll out.
Being introduced to several hospital settings at the beginning of each rotation presented the fellows with the additional tasks of quickly familiarizing themselves with and acclimatizing to new organizational rules and procedures, staff and institutional cultures.
If you need to quickly re-familiarize yourself with Australia, you might consider watching the turn-of-the-century classic "Our Lips Are Sealed".
High-school educated staff were able to quickly learn and familiarize themselves with the tablet and data collection software.
He can familiarize himself quickly with the subject and develop the idea into an opportunity without any great assistance.
Learn how to quickly reload, unjam, draw and familiarize yourself with the correct type of ammo.
In order to better understand the heterogeneous array of reactions to a death and the importance of posting quickly about it, new members should first familiarize themselves with Elisabeth Kübler-Ross's Five Stages of Grief.
