It's an interesting discussion and one that I hope that the other commentor's won't dismiss out of hand. The main hypothesis is: Does having a larger social following increase a brand's legitimacy/authenticity/trust with a consumer?
Unfortunately, there are not any case studies that I can see that test this theory, and certainly not any that bought Facebook likes outright. As you'd expect, there are a number of sources that say it's a bad idea, such as this, and this.
On a complete hunch, I'd guess that it would help increase your perceived "trust" with a potential client, but due to a lack of evidence I can't recommend you doing so - in this way at least.
I would, however, encourage you to promote your Facebook page to increase your likes and followers with targeted and engaged users. You can do this through promoted advertising on Facebook itself, or other places like Twitter or reddit. It will cost more to reach a figure of, say, 1000 likes than just outright buying them - but this way you'll know that the people liking your page, if targeted correctly, will be of a relevant audience and much more likely to engage with your brand. If you have a Facebook page bursting with user activity, that will certainly help to build trust too.
I suppose it comes down to like most things in life, if you cut corners you risk doing more harm than good. Promoting your Facebook page in a proper way will not only increase likes (and potentially perceived trust), but in doing so you are introducing your brand to your target audience - meaning you could get conversions out of it you might not have done before.
These links will help you out:
How to Use Promoted Page Ads to Get Tons of New Likes on Facebook
A Beginner’s Guide to Facebook Ads
The Ultimate Guide to Running Online Competitions
As for testing whether just buying likes increases "trust", it would be an interesting study to see that's for sure.
Hope this helps.