Thank you for asking this. I use this plugin a lot, so I had a look to see if it could be automated (as I tend to always ask for similar things like highways and buildings). It turns out that several of the plugin's algorithms have been made available through Processing. This will make things much easier - clipping isn't difficult so it never occurred to me to model it!
If you want to clip to an arbitrary polygon (like clipping your roads to the outline of a state) you can do this with a processing model
Here, I asked for a small diamond area and all the buildings in it, you'll see some of the buildings have been clipped

Model looks like this.... (I just processed the multipolygons, but you can add to this model to process other geometries easily).
There's a 'save as image' option on the graphical modeller, makes screenshots a lot easier...

And here's the output after pressing the 'save as python script' button. You can generate the XML using the "generate query" button using the dialog GUI. This query asks for buildings (lines & polygons only)
##quickosm1=name
##clipoutline=vector
##extent=extent
##result=output vector
outputs_QUICKOSMQUERYOVERPASSAPIWITHASTRING_1=processing.runalg('quickosm:queryoverpassapiwithastring', 'http://overpass-api.de/api/','<osm-script output="xml" timeout="25">\n <union>\n <query type="way">\n <has-kv k="building"/>\n <bbox-query {{bbox}}/>\n </query>\n <query type="relation">\n <has-kv k="building"/>\n <bbox-query {{bbox}}/>\n </query>\n </union>\n <union>\n <item/>\n <recurse type="down"/>\n </union>\n <print mode="body"/>\n</osm-script>',extent,None,None)
outputs_QUICKOSMOGRDEFAULT_1=processing.runalg('quickosm:ogrdefault', outputs_QUICKOSMQUERYOVERPASSAPIWITHASTRING_1['OUTPUT_FILE'],None,None,None,None,None,None,None,None)
outputs_GDALOGRCLIPVECTORSBYPOLYGON_1=processing.runalg('gdalogr:clipvectorsbypolygon', outputs_QUICKOSMOGRDEFAULT_1['multipolygons_LAYER'],clipoutline,None,result)
Tip: if you're not working in 4326, switch to 4326 in OTF. That way, when you choose the layer extents when running the model, you'll get latitude and longitude extents. Otherwise you'll get the "mismatched CRS" error.