In most games and situations, no. In-game trade Pokemon are usually fixed and preset, so they're shiny value will always be set as false, or not shiny.
There are a couple situations where this is/may not be the case.
In RBY, in-game trade Pokemon have random DVs, so when traded up to Bank or GSC, they can potentially be shiny.
In-game trades in Pokemon XD and LGPE are not shiny locked, and any Pokemon obtained from an in-game trade has a 1/8192 or 1/4096 chance respectively of being shiny. In LGPE, these odds can be boosted to 1/1365 with the Shiny Charm
Trades with Regina in the Isle of Armor DLC have randomized values and can be shiny.
It is also possible that Yancy/Curtis trades from B2W2 are not shiny locked. As the genders and natures of Pokemon received in this fashion are random, it is also possible that personality values and therefore shininess of the Pokemon are random also. Unfortunately there is no evidence or account of someone obtaining a shiny through these means anywhere on the internet, so we currently have no way to prove this. If they are not shiny locked, their shiny odds would be 1/8192.
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