Can Bitcoin-Qt be configured to trim the blockchain?
Since Bitcoin-Core 0.11.0 you can prune (trim) the blockchain in Bitcoin-Qt.
But not from the UI.
You need to add prune=550 to your bitcoin.conf file and restart Bitcoin-Qt.

Explanation:
-prune=&lt;target in MiB&gt; will tell bitcoin-core to remove blocks which are older than oldest block that can be kept with a chainsize (sum of block-sizes) of &lt;target&gt; (min. 550MiB as example above shows). A minimum of 550MiB ensures to keep at least 288 blocks (~2 days). 
(simple explanation not including undo-data)

Restriction:
-prune does disable all wallet functions in version 0.11 (will work with a wallet in version 0.12).