Appear Under Google Suggested Keywords

Is that any service available in the market where can make my brand name appear when searching for relevant keywords in Google Search Engine?

Example when typing:
dog for sales (then my brand name will appear after the organic keyword)

You will need to build your brand around that topic, topical relevance is the key here.
I don't know any specific services for achieving that, but you will need to hire someone who's an experienced in SEO and branding.
Quality and only niche relevant backlinks will help you with that.

If you want your brand to appear for respective keyword, optimize the brand for that keyword.
Increase the brand awareness, make it popular, promote it on various platforms.
I would take time, once it appears then better you try to maintain it.

For service or to hire someone check Hire a Freelancer section https://www.blackhatworld.com/forums/hire-a-freelancer.76/

Interested one will contact you.