I can think of a couple of possibilities here. There are probably others.
One is that the software you're using to calculate p-values is applying something like a Bonferroni correction. This XKCD cartoon illustrates the problem nicely: the more statistics you look at, the greater the probability of finding a "significant" result purely by chance.
Bonferroni attempts to mitigate this by adjusting the threshold for significance according to the number of things you're checking for significance. In the first case, you have two parameters; in the second, you have three. So a value that appears "significant" when you have two parameters may no longer meet the tougher threshold that applies when you have three.
Another possibility is that your data can be described either as one weak linear effect, or as a combination of a weaker linear effect with an also-weak quadratic effect.
For example, say my data is clustered around (0,0) and (1,1). In a linear model, my best-fit approximation is y=x. In a quadratic model, I could end up with y=x, or with y=0.5x+0.5x^2, or various other possibilities; small differences in the inputs could make a large difference to the model parameters.
In this case, if you're replacing a weak linear effect with a weaker linear effect + a weak quadratic effect, you can end up dropping the "significance" of the linear effect a little bit below the cutoff.
If your data is such that x and x^2 are close to linearly dependent, you can expect poor behaviour generally.
Even small changes to a model can cause a p-value of 0.049 to change to 0.051 or vice versa. If we arbitrarily choose 0.05 as our cutoff, yes, that can cause a result to change from "significant" to "not significant".