Placing the adverb 'quickly' before or at the end of a verb phrase can express the sense that time was not wasted whereas placing it after the verb expresses the manner in which the action occurred. You can't see this difference very well in your example, so I'll change it.
After dinner, she quickly finished her homework and sat down on the couch, ready to start the film.
The idea is that she didn't want to waste time. She wanted to keep the time spent doing her homework and the time between dinner and watching the film as short as possible. It doesn't necessarily mean that she did her homework any faster than usual. The adverb 'quickly' is usually used with this meaning. It doesn't make an important difference whether you place it directly before or directly after the verb phrase:
After dinner, she finished her homework quickly and sat down on the couch, ready to begin.
If you want to use 'quickly' to mean 'at high speed', it's better off in the manner position, which is directly after the action verb that it modifies. Let's use 'fast', which has the same meaning of high speed that you want, to see this a bit more clearly:
She ran fast.
That's verb + adverb. Good.
It's not right to put a manner adverb before the verb (usually):
*She fast ran.
That's adverb + verb, which is wrong.
So, if you mean that she ran at high speed (which I guess you do), then say She ran quickly.
