Q:

Are our minds quantum entangled with people we are (emotionally) close to?

I came to know about the extended mind from the following presentation by Rupert Sheldrake who presented his research at Google TechTalk. 
The link to the video
Sheldrake claims that the horizon of power of our brain is not limited to the physical brain which we see and observe but its extended outside also. This is like a invisible radius of power like electromagnetic field; when we are very close to some person, we can feel and get sense of their action, no matter what is the distance they are away. 
Just after the 18m mark he states:

Another way of thinking about this is through quantum entanglement and Dean Radin who gave a talk here at Google some time ago -a few months ago  - has written a book called Entangled Minds suggesting that when people have close social relationships and interactions their minds become entangled so that when they separate there is a still an entanglement between them and so a change in one is reflected by a change in another. 
Quantum non-locality - entanglement of quantum particles - is well known, well documented and is in fact the basis of now 
  technological applications in quantum cryptography and quantum computing. And what that shows is that particles that are part of the same system when they move apart, a change in one is immediately associated with a change in the other. This is independent of distance unlike many other physical phenomena. It doesn't matter if it is an inch apart or a mile apart; it's distance independent entanglement or non-local connection.
I am suggesting something like that happens in the case of animal groups, and in social groups in general, including human groups.

Do our minds get quantum entangled?

A:

No one knows what "mind" is, but the brain is a bad candidate for hosting quantum superpositions, much less sharing them with other people.
The most credible proponent of mind having a quantum component is the physicist Roger Penrose who has argued, beginning in his 1989 book The Emperor's New Mind that the human mind displays non-computable behavior and thus cannot be a Turing machine. He proposed quantum phenomena as a route towards non-algorithmic computation. 
Penrose's logic by which he makes the conclusion that the brain is not a Turing machine has been widely criticized, and Wikipedia quotes Princeton's John Burgess as saying:

the consensus view of logicians today seems to be that the Lucas-Penrose argument is fallacious, though as I have said elsewhere, there is at least this much to be said for Lucas and Penrose, that logicians are not unanimously agreed as to where precisely the fallacy in their argument lies. There are at least three points at which the argument may be attacked.

In 1996, Stuart Hameroff and Penrose published Orchestrated Objective Reduction of Quantum Coherence in Brain Microtubules: The "Orch OR" Model for Consciousness, which proposed that the physical structure called "microtubules" in the brain might be a mechanism by which quantum effects could affect neurons, whose structure is otherwise firmly in the world of classical physics. In that text, they acknowledge that:

Warm, wet and noisy, the brain at first glance seems a hostile environment for delicate quantum phenomena which generally demand isolation and cold stillness (superconductors), or energy pumping of crystals (lasers). 

The "Orch OR" theory has been heavily criticized. In Falsifications of Hameroff-Penrose Orch OR Model of Consciousness and Novel Avenues for Development of Quantum Mind Theory, Georgiev provides: 

a list of twenty four problems not being repaired for a whole decade after the birth of the model.

In other words:

Most philosophers don't see the need for a quantum component of mind; and
There's no compelling theory for a quantum component of mind

If there is a quantum component of mind, we don't know how that component works. We certainly don't know how it interacts with other minds. 

A:

Do our minds get quantum entangled?
Just need to know if there is really any mathematical model or in physics it is approved?

Wikipedia's Quantum mysticism article suggests that the answer is "no".
Wikipedia is also critical of the author you quoted, Rupert Sheldrake, saying,

Sheldrake's "morphic resonance" hypothesis is widely rejected within the scientific community[5][6][7][8] with some calling it pseudoscience[8][9][10][11][12] and magical thinking.[9][13] Concerns include the inconsistency of the hypothesis with accepted scientific theories and associated evidence for them,[8] lack of evidence for the hypothesis,[14][15][16][17] and concern that the hypothesis is overly vague[9][10] and unfalsifiable.[9][15] Sheldrake's experimental methods have been criticised for being poorly designed and subject to experimenter bias,[18][19][20] and his analyses of results have also drawn criticism.[10][21]
In promoting his hypothesis Sheldrake occasionally garners public attention through television and other media outlets, and has been accused of negatively impacting the public understanding of science.[9][10][14]

He is a former biochemist, not a physicist.
The video excerpt which is cited in the OP references a book titled Entangled Minds whose blurbs (i.e. promotional reviews) are not written by physicists. The first blurb is by Deepak Chopra who at timeline 3:10 in this video says that physicists have "somehow hijacked the word for their own use".
IMO the term "quantum", as and when it is used by 'these people', doesn't have the same meaning as when it's used by most physicists and/or applied mathematicians.
I accept that this answer of mine is "ad hominem": i.e. it concentrates on the people more than on the argument. On the other hand, if you're looking for "peer-reviewed" literature, this answer may be useful to you, in that identifies who they regard as they own peers.

