I am new to blender. Pressing ⎈ Ctrl⎇ AltQ opens the quad view. How to maximize any one of the four views of the quad view?
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To maximize a view you can hit Ctrl + up arrow, to get back to normal use Ctrl + down arrow. Make sure the cursor is in the view you want to maximize.
Most of the time I just use the numpad hotkeys gandalf3 mentioned though.
As I had forgotten and Gandalf3 noted in the comments, this does not work for the actual "quad view" in a single viewing window. Instead of using the Ctrl. + Alt. + Q quad view, I usually just set up a manual quad view using multiple viewing windows by dragging the little slanted lines at the top right corner of the view. Then the Ctrl + up arrow/down arrow will work. You can make preset views by clicking the "+" at the top of the window (where it says "Default" now).
The quad view exists so that you can see all the views at once.
Instead of thinking of it like maximizing one of the quad view windows, think of it like aligning the user controlled view to an axis.
Holding ⎈ Ctrl will snap to the opposite direction of the axis, so ⎈ CtrlNumpad 7 snaps to bottom view.
Numpad 5 toggles perspective/orthographic view.
Also see the wiki for more information on navigating the 3D view.