I have two variables which may have a nonlinear relationship, according to the descriptive statistics.
The independent variable has a positive and significant coefficient in the OLS regression. When a square term is included in the model this becomes negative and insignificant while the square term becomes positive and significant.
Can I argue that these variables have a nonlinear relationship?
You need to be careful with the way you phrase this question. And, in general it would be helpful to see the output of the tables you are describing.
If I understand you correctly, you are saying that you fit models of the form
Model A: $\quad y = b_0 + b_1 x + e$
Model B: $\quad y = b_0 + b_1 x + b_2 x^2 + e$
where $y$ is the dependent (response) variable, $x$ is the independent (explanatory) variable and $e$ is the error term (residuals unexplained by the model) and the $b_i$ ($i \in \{0,1,2\}$) are coefficients.
Both A and B are linear models because the coefficients appear as additive terms. From what you say, under Model A, the $b_1$ term is significant and positive. Under Model B the $b_2$ term is significant but not $b_1$.
To preserve marginality the $b_1$ term must be retained in Model B when $b_2$ is significant, even if $b_1$ itself reports as non-significant. You can think of this informally as: you can't have $x^2$ unless you already have $x$. That result is telling you that inclusion of the $x^2$ term provides a better fit to the data than the model without the $x^2$ term. This indicates that curvature is a feature of the relationship between $x$ and $y$. In that sense the relationship is non-linear. But the fitted model is still a linear regression model, albeit one which allows a degree of curvature to be accommodated via a polynomial term.
