My first impression of a site is always based on the design for sure. If the site looks interesting or unique visually, I'm more likely to stick around and see what's there.
Good design absolutely can build trust between users and a company. I think expressing your brand's personality or point of view in your design choices is essential to creating any kind of emotional connection or being memorable for your audience.
For me personally, this is especially true when a design looks really dated. It's hard not to feel a bit uneasy when going through a checkout process that looks like it was designed10+ years ago and hasn't been touched since!