Not sure if I understand your question entirely, but an antioxidant is a reducing agent that acts as a "decoy" in a sense for free radicals to attack. The oxidizing radicals steal electrons from the antioxidant, thereby neutralizing them, and those electrons can then be replenished relatively easily. Antioxidants should, in theory, slow the browning process.

Edit: it could also be the antioxidant that "attacks" the radical. This probably varies with the specific compound(s).

Edit 2: My answer now sounds like the health 101 explanation. Based on the things I have learned from other peoples' answers, browning and antioxidant activity are related, but not dependent on one another. The reason that processed foods do not brown quickly is because there are synthetic antioxidants added (i.e. mixed tocopherols and things like that), so I would guess that these are more active as antioxidants, though they are (maybe counter-intuitively) less healthy for you because the compounds that are attributed to this effect are much less diverse, not to mention synthetic (meaning they don't fit in the biological lock-and-key mechanisms as well as natural antioxidants).

Generally, phenolic compounds and ascorbic acid (Vitamin C) confer a degree of antioxidant activity in fruits and vegetables. This obviously depends in great part on the quantification method you use. For instance, many studies on apples (Eberhardt et al., 2000; Schmitz Eiberger et al., 2003; Sacchetti et al., 2008) have revealed that antioxidant activity (Trolox) tend to correlate with polyphenol content but not with ascorbic acid (a well-known antioxidant with strong effect against enzymatic browning). So I wouldn't consider the degree of browning as a reliable indicator for antioxidant activity.

There could be multiple reasons for a fruit turning brown faster than another. In principle, the phenolic compounds and polyphenol oxidase (PPO) are originally separated but a bite (e.g. apple) causes cell disruption, mixing of compounds and exposure to oxygen; this eventually leads to the formation of e.g. melanines that confer brown colour to the fruit.

This process could be accelerated if the content of PPO is high and/or the content of antioxidants, especially ascorbic acid, is low (type of fruit, variety, season etc.). Enzymatic browing can also be caused by high temperature and/or high relative humidity during storage. The possibilities are endless if you also consider processed products.

All this will definitely affect the antioxidant activity in the product. But, chemically, antioxidant activity will continue after the colour of the fruit has shown signs of browning.

If the question is: Fruits and vegetables that quickly turn brown after cutting or mincing can be considered foods with polyphenols? the answer is yes. You cannot have false positive. If turning brown, a fruit or a vegetable necessarily contains polyphenols, but not only because it also means that it contains polyphenol oxidase (PPO), and that its content of vitamin C and its availability at the site of action is low.

The next question could be: Are polyphenols necessarily antioxidant? The answer is yes apart some rare exceptions.

So, we can answer your question by saying that indeed fruits and vegetables that quickly turn brown after cutting or mincing can be considered foods with antioxidant activity for the reason that they contain polyphenols. However, you cannot infer that such fruits have an antioxidant activity higher than that of a fruit without any browning. It may just contain more PPO.

Enzymatic browning just tells you if polyphenols are present or absent in a qualitative way. It does not tell you how much.

It is not the antioxidant property, it is the enzymatic browning or oxidative browning that mainly occurs by oxidase and peroxidase like enzymes and by oxidation of cellular compounds that are present in the fruits and vegetables. When we cut fruit or vegetable the cellular components (not always phenolics) rapidly gets oxidized and appears brown.

My doubt was: if the fruit or vegetable is quickly browned, at the time to consume it will have lost its antioxidant activity.

You rightly point out that, despite the browning of polyphenols, other antioxidants may remain active as they are less susceptible to oxidation.

I also think that just the browning of polyphenols protects against oxidation other antioxidants extending their activities.

Also, only the cells that have been broken will have suffered the browning reaction, so the cells below the surface won't be affected, and will still contain unreacted antioxidants.

To my understanding, less the amount of antioxidant in fruit or vegetables more quickly it will turn brown on keeping in air

Browning and antioxidant activity are related, but not dependent on one another. These are free radicals which react with the metal, and metalic oxides due to exposure of air darken the color.