Is the sentence "we value" correct? As in the meaning of "we matter"?

No. "Value" is a transitive verb, so it requires an object. You can't simply say "We value." What's more, if you did, it makes no suggestion that whom "we value" is ourselves.

"Matter" in that sentence is intransitive. It requires no object. It is also reflexive, so the subject of the verb "we" is also the object, meaning it's "we" that matters.

"Value" is not reflexive and makes no intimation. Simply saying "we value" will make people wonder what it is "we value." No one will assume that it's suggesting "we value ourselves."

To use "value" in a sentence that means "we matter," you'd have to use it as a noun (i.e., "We have value.").

No that would be "we are valued"

No. "We value" would be followed by a noun- the thing that we value. "We value good manners" for example.
"we matter" is more of a passive construction- "We matter to others and ourselves."
"Good manners matter to us."

No...you must value 'something' as in "We value democracy".

We value is NOT at sentence. It is a phrase. a simple phrase consisting of a subject (we) and a verb (value). It indicates that you and someone else or a few someones see value, worth, importance, etc in something like a person, object, idea, etc. They may include a name of the person who you value, the object like a cell phone, or an idea or concept like freedom. We matter is also a phrase, but it suggests you (and whomever) are important or of significance to what or whomever. To value, or consider important, what or whomever is not the same as mattering or being of consequence. Since these are phrases and not complete sentences, there is no way to tell what they actually mean, especially without context or the related information that would indicate what it is you are trying to convey.