No, it's not correct.

The verb mind can take an Equi Gerund Complement:

• Would you mind doing something?

but not an Equi Infinitive Complement:

• *Would you mind to do something?

whence the asterisk on the second sentence, indicating that it's ungrammatical.

Verbs vary a lot in which complements (if any) they can take; this is one part of the meaning of the verb. Like and care are both verbs that can take infinitive complements; mind isn't. That's all, really.

Any good ESL dictionary should indicate which types of complement a verb can take; dictionaries for native speakers, on the other hand, usually don't mention that. One more reason not to look in dictionaries for grammatical information.

No, it's not correct.

The verb mind can take an Equi Gerund Complement clause:

• Would you mind doing something?

but not an Equi Infinitive Complement clause:

• *Would you mind to do something?

whence the asterisk on the second sentence, indicating that it's ungrammatical.

Verbs vary a lot in which complement clauses (if any) they can take; this is one part of the meaning of the verb. Like and care are both verbs that can take infinitive complement clauses; mind isn't. That's all, really.

Any good ESL dictionary should indicate which types of complement clause a verb can take; dictionaries for native speakers, on the other hand, usually don't mention that. One more reason not to look in dictionaries for grammatical information.