Would you mind opening the window?

to say Would you mind+... in japanese is easy, you can use the following phrase

...mo yoroshii desuka?　。。。＋も　よろしいですか？
... mo ii desuka? 。。。＋も　いいですか？

this some example for those phrase :

1. Would you mind opening the window?
まど　を　あけても　よろしいですか？　Mado wo aketemo yoroshii desuka
2. Would you mind i read this book ?
この本　を　よんでも　いいですか？ kono hon wo yonde mo ii desuka ?
3. Would you mind i use this ballpoint ?
この　ボルーペン を　使っても　よろしいですか？ kono boru-pen wo tsukatte mo yoroshii desuka
4. Would you mind i take a picture
者品　を　とっても　いいですか？ - syashin wo totte mo ii desuka

it's easy right !

mado : window
akeru : open
(ake-nai, ake-masu,akeru,akeru-toki,akereba,akero,akeyou,aketa)
yoroshii / ii : good, ok
mado o akeru : open the window
hon : book
yobu : read
(yobanai, yobimasu, yomereba, yomero, yonda, yonde)