Find the smallest rhythmic value used in your piece and "scale everything up" so that smallest value in instead the smallest value supported by the game (so, if the smallest value is the sixteenth notes -- the double-stemmed notes -- in your screenshot, and the game only supports quarter-notes, then you need to quadruple the tempo and "multiply" all your rhythmic values by four; alternatively, if you were mistakenly referring to the sixteenths as eighths, then you might only need to double everything).
I find it hard to believe the game doesn't support rests in any way, can you seriously only place notes directly adjacent to each other? If the answer to that is "no, you can actually place notes non-consecutively", well then, you have rests, they're just not reflected in notation. If the answer is "yes, seriously, it's very strange", then you have two-maybe-three options: (1) just, don't use rests, let the last note you played sustain until the next one sounds; you'll get some new dissonances doing this but they'll probably resolve just fine. (2) take some creative liberties, do something new in those silences. (3) if the game supports multiple lines at once, mask the issue and simulate rests by putting the "resting" line in unison with another lines.
That said, without actually seeing the game first-hand, or having a precise and formalized description of the game's features/capabilities, there's no way of knowing whether this answer is really correct/the best way of doing things.