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I have a domain that has external links to its sub-URLs like example.com/a or example.com/b
Now I want to use this domain and redirect all those sub-URLs to root URLs (example.com) because my website is one pager. Will the root URL get extra quality from these back links through the old sub-URLs?
A redirect to a dissimilar page will not pass any link juice (SEO value).
In your case, you have mentioned that you are going to redirect several URLs (naturally different from the homepage) to the homepage. Hence you cannot expect link juice.
Google does not like redirects to the home page. It treats redirects to the home page exactly like the page returned a "404 Not Found" status. Google calls such redirects "soft 404" errors. Redirect to the home page appear in the soft 404 report in Google Search Console.
Google documents this practice at https://support.google.com/webmasters/answer/2445990?hl=en
Google treats 410s (Gone) the same as 404s (Not found). Returning a code other than 404 or 410 for a non-existent page (or redirecting users to another page, such as the homepage, instead of returning a 404) can be problematic. Such pages are called soft 404s, and can be confusing to both users and search engines.
If you want to capture SEO value from these external links you will need keep their old content on their old URL, or on some other deep URL to which you can redirect without triggering soft-404 errors to Google.
