when finding the p-value of a chi-square test,

That really depends on two things:
1. what hypothesis is being tested. If you're testing variance of normal data against a specified value, it's quite possible to be dealing with the upper or lower tails of the chi-square (one-tailed), or both tails of the distribution. We have to remember that (O-E)2E(O-E)2Etype statistics are not the only chi-square tests in town!
2. whether people are talking about the alternative hypothesisbeing one- or two-sided (because some people use 'two-tailed' to refer to a two-sided alternative, irrespective of what happens with the sampling distribution of the statistic. This can sometimes be confusing. So for example, if we're looking at a two-sample proportions test, someone might in the null write that the two proportions are equal and in the alternative write that π1≠π2π1≠π2 and then speak of it as 'two-tailed', but test it using a chi-square rather than a z-test, and so only look at the upper tail of the distribution of the test statistic (so it's two tailed in terms of the distribution of the difference in sample proportions, but one tailed in terms of the distribution of the chi-square statistic obtained from that -- in much the same way that if you make your t-test statistc |T||T|, you're only looking at one tail in the distribution of |T||T|).
The chi-squared test is essentially always a one-sided test. Here is a loose way to think about it: the chi-squared test is basically a 'goodness of fit' test. Sometimes it is explicitly referred to as such, but even when it's not, it is still often in essence a goodness of fit. For example, the chi-squared test of independence on a 2 x 2 frequency table is (sort of) a test of goodness of fit of the first row (column) to the distribution specified by the second row (column), and vice versa, simultaneously. Thus, when the realized chi-squared value is way out on the right tail of it's distribution, it indicates a poor fit, and if it is far enough, relative to some pre-specified threshold, we might conclude that it is so poor that we don't believe the data are from that reference distribution.
If we were to use the chi-squared test as a two-sided test, we would also be worried if the statistic were too far into the left side of the chi-squared distribution. This would mean that we are worried the fit might be too good. This is simply not something we are typically worried about.
Results:
Based above description we concluded that statement is false.
chi-square test is essentially good fit to the one tailed data compared to two tailed data.