Yes. Anyone who has the technical skill can translate material. Translation is a skill that has to do with understanding the technique and fluency in the languages/cultures, and it's certainly possible for non-Christians to have the requisite skills, just as it's possible for a Christian not to have it. That should be fairly obvious, so I assume by "trust" you are wondering about them getting away with altering the texts or getting them wrong.
If that's the case, let's step back a moment so I can ask how you'd go about verifying if a Christian got it right. You wouldn't know for sure if anyone, Christian or not, got it right, unless the work had undergone some form of peer review]1, where other translators verify the work. This is standard and common in translation, history, and any discipline where someone can make a mistake.
The risk of a Christian and non-Christian making errors in understanding the meaning is the same, so in one sense, the answer is "no", but not because they're not Christian, but because they are human, and therefore not perfect.
We can grant a non-Christan the same amount of trust that we'd grant a Christian simply because peer review for catching errors is an implicit part of establishing whether the work was done right, regardless of who did the work.
It's the same principle as the one behind manuscript evidence. The more examples to compare, and agreement we have, the higher confidence we can have that we have it right.
There would be so many versions to compare it to that obvious changes that affect doctrinal issues would be discovered and lambasted immediately after it was published. Not to mention the hundreds of experts that could refute and dispute errors, intentional or otherwise.