If you say 'Right of education', 'Right of liberty' or 'Right of property' people will probably understand what you mean, but you will sound like a beginning speaker of English. These phrases aren't used by native speakers.
The confusion is understandable given the examples you've shown, however you're assuming there's a rule in English that there actually isn't. The prepositions 'to' and 'of' can't be interchanged in different phrases like you've suggested just based on the parts of speech (nouns or verbs) that come after them.
English is a language of phrases and idioms that have to be learned by rote in many cases, unfortunately for the learner. You can say
Need of food
Need of education
Need to read
Right to food
Right to education
Right to read
Right of food
Right of education
However, you can say
Right of way
which is the only phrase I can think of that uses 'Right of' with a noun, meaning a right to the thing named by the noun. You can say
The right of people
The rights of all Americans
but in this case we are still saying 'the right ... to something' as in
The right of all people to food and shelter
If you want to learn native English, I would encourage you to take your examples from known native speaking sources - movies from native English speaking countries or books written by native English speaking authors. If you live in a place where a lot of people are speaking English as a second language you will hear examples of non-native English that have been widely adopted and are commonly used even though they are 'incorrect' from the standpoint of a native speaker. For instance, as mentioned in the comments, the phrase 'will of car' has no meaning in native English.