Can I say “montrer quelque chose quelqu’un”

No, you say :

> Montrer quelque chose à quelqu'un

Example :

Je vais montrer ma nouvelle voiture à James.

And what abour persons, can I say “te monter à moi” or “me te monter”?

No. Here, you say (continue with the same example as above) :

Je vais te montrer ma nouvelle voiture.

In this sentence, I’m going to show my new car to James, represented by te.

And, James will say after seeing my new car :

John m'a montré sa nouvelle voiture

Intuitively the difference seems clear but I had trouble understanding exactly how it works. Here is what I could figure out.

To show somebody something

Montrer quelque chose quelqu’un

is agrammatical, you need to use the preposition “à”, as in your first example

Montrer quelque chose à quelqu’un

Talking about persons

In this case, there is actually a difference between direct and indirect object. When you are talking about two different persons, “me” necessarily refers to the direct object (i.e. I am the person you are showing), whereas “à moi” is indirect (i.e. I am the person to whom you are showing someone).

So

Te montrer à moi

is correct, whereas

me te montrer

is not. You could however write

Me montrer à toi

Things and reflexive pronouns

Reflexive pronouns like “me” and “te” can also refer to an indirect object, but only if there is also a thing as a direct object in the sentence and you are not using “à […]”, as in

Me montrer sa voiture.

In this case,

Montrer sa voiture à moi

is understandable (and something children might say) but incorrect.

Reflexive use

One thing that gets lost with the infinitive is that “montrer” can also be used reflexively, as in:

Je me suis montré [à toi]

In particular, “je me suis montré” can imply something like “I showed up so that people notice that I am here”.

But the principle identified before remains valid, when there is no other object, “me” is the direct, not the indirect object.

Usage

Note that while these constructions are grammatically correct and probably understandable, using “montrer” often sounds contrived in these examples. In many cases you would use “désigner” ou “présenter” instead.

You can also convey related ideas in a different way with a relative proposition, e.g.

Il m’a montré qui tu étais.

While the meaning is very different, the same principles are at play with “donner”.

Tu te donnes à moi

Je me donnes à toi

Je te donne un crayon

Another source of confusion

Confusingly, you will also encounter

Me montrer quelque chose à moi

where both “me” and “à moi” refer to the same person, the indirect object. It’s a colloquial (possibly somewhat childish) way to mark emphasis (as in “I have seen it but you haven’t”).

Thinking about this further

I think the most important clue to interpret these sentences correctly is to compare the subject, the reflexive pronoun and the indirect object. If they are all different, it’s clear which is which. If you use the same person, you will get either a different meaning (“Je me […]”) or emphasis (“me […] à moi”).

The construction of this sentence is less flexible in French than in English. The construction depends only on whether pronouns are used or not, there’s no choice.

The normal construction is: subject, verb, direct complement, indirect complement.

Elle montre la maison à son copain.   (She shows the house to her (boy)friend.)

If the direct complement is a pronoun, it is placed before the verb.

Elle la montre à son copain.   (She shows it to her (boy)friend.)
Elle te montre à son copain.   (She shows you to her (boy)friend.)

If the indirect complement is a personal pronoun introduced by the preposition à, it is placed before the verb and the preposition disappears.

Elle lui montre la maison.   (She shows him/her the house.)
Elle me montre la maison.   (She shows me the house.)

When both complements are pronouns, things get complicated. If both complements are third-person pronouns, the direct complement comes first.

Elle la lui montre.   (She shows it to him.)

But if the direct complement is third-person and the indirect complement is first- or second-person, the indirect complement comes first.

Elle me la montre.   (She shows it to me.)

And if the direct complement is first-person or second-person, then the indirect complement is always placed after the verb, with its usual preposition, even if it is a pronoun.

Elle te montre à moi.   (She shows you to me.)
Elle nous montre à toi.   (She shows us to you.)

By the way, you’ll notice that the form of the pronoun changes depending on the construction:

• First/second person singular: normally moi/toi, but me/te before the verb (whether it’s a direct or indirect complement).
• First/second person plural: nous/vous (no variation).
• Third person: normally lui/elle/eux/elles, but le/la/les/les for a direct complement placed before the verb and lui/lui/leur/leur (no gender variation) for an indirect complement placed before the verb.