If Pearson's correlation is zero does this imply no linear correlation? I am looking to detect linearity in a dataset. Linearity as in the linearity assumption of linear regression. (There exists a linear relationship between the independent variable, x, and the dependent variable, y. From here)
And I had the idea that if Pearson's correlation can only be applied to linear relationships of variables... couldn't I use Pearson's correlation and if it is near zero, that means the relationship between the two variables is NOT linear?
I know that they could non-linearly correlated or not at all, but that is not what I want to know.

Your interpretation of Pearson correlation, as measuring the linear relationship between variables, is accurate.  However, you need to be a little careful in making the distinction between the true properties of the relationship between variables and an estimator pertaining to those properties.  You also need to go further than just referring to the Pearson correlation as "low" without elaborating on how we can tell that "low" is.  I will describe how these two issues are dealt with below.
We generally draw a distinction between the true correlation coefficient and the sample correlation, with the latter being an estimate of the former.  Pearson correlation is a form of sample correlation and constitutes an estimator of the true correlation.  In the code below I use the cor.test function to conduct a hypothesis test for correlation, using the Pearson correlation as the test statistic.
#Generate some mock data (uncorrelated)
X <- rnorm(100)
Y <- rnorm(100)

#Conduct permutation test for zero correlation
set.seed(1)
TEST <- cor.test(x = X, y = Y, alternative = 'two.sided',
                 method = 'pearson', exact = TRUE, conf.level = 0.95)

#Show results of test
TEST

        Pearson's product-moment correlation

data:  X and Y
t = 0.44934, df = 98, p-value = 0.6542
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.1524321  0.2396276
sample estimates:
       cor 
0.04534364

In this case we have an observed Pearson correlation of $r = 0.04534$ and the p-value for the test is $p = 0.6542$.  Based on this test we do not reject the null hypothesis that the variables are uncorrelated ---i.e., have no linear relationship.
This type of test is what allows us to say that the magnitude of the Pearson correlation is sufficiently "low" that it indicates no linear relationship.  Bear in mind that "low" is relative to sample size, so it is not a fixed threshold.  With these elaborations in mind, your essential idea is correct.
