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.

Well, yes, if a Pearson correlation is near zero, then by definition with the assumptions of linearity in Pearson correlations there is no linear relationship. As an example, here is a pair of random sampled variables with no relationship with each other:
x.non <- rnorm(n=1000)
y.non <- rnorm(n=1000)

When plotting them and adding a regression line:
plot(x.non,
     y.non)
abline(lm(y.non ~ x.non),
       col="red")

We get this, a sprinkling of very random data points with no real structure and a flat line, which we don't want.:

I haven't used a seed, but checking the correlation:
cor(x.non,
    y.non)

It should be near zero like below:
[1] -0.02072745

Now if you created a conditional relationship where y is dependent on x, you should have a strong relationship, as shown below:
x.cor <- rnorm(n=1000)
y.cor <- x.cor*2 + rnorm(n=1000)

plot(x.cor,
     y.cor)
abline(lm(y.cor ~ x.cor),
       col="red")


You can quite clearly see the line runs diagonal, not flat. Your data points should also resemble a line like the plot shows, rather than the blob of data points shown earlier with the random data. This is what is meant by linearity: your dots should look as close to a line as possible to hold the linearity assumption. Checking the correlation:
cor(x.cor,
    y.cor)

It should be fairly strong:
[1] 0.8892965

Here is a comparison of linearity and the data's corresponding Pearson correlation value for four examples to compare, ranging from no linearity/no correlation to perfect linearity/perfect correlation:

Hope that answers your question and let me know if you need further clarification.
