Python program that checks connections based on URLLIB response
It's a neat idea, I like it! There is however a couple different things that could be done differently, while retaining the original idea.

Syntax

Have a look at PEP-8, it's a universal style guide for Python, it will give you an idea of how to style your code. As an example: functions, methods and variables should be snake_case, not mixedCase or CamelCase. While the use of the latter won't raise a syntax error, using PEP-8 is preferred.

Try to avoid wildcard imports (importing contents of entire libraries), as it will pollute your namespace. In the point below I suggest a library that's not part of the standard library, which should be separated from standard library imports, by one line.

import tkinter as tk

import requests


attemptConnect function

I would consider switching to a higher level web request solution, the requests library. It comes with all the features we need out of the box and additionally it provides an extremely handy Session object. Basically, it will use the underlying TCP connection making the subsequent requests faster. We will send a HEAD request, so the content won't be requested.

At first I thought the global variable is a counter of some sort, but a closer look made me realise it acts as a boolean variable. Why not make it one? This change will also have an effect in your ternary: "#38f789" if request_successful else "#ff6656".

I've taken the liberty to start changing function and variable names to follow PEP-8.

session = requests.Session()
...
def verify_url_status():
    global request_successful
    try:
        session.head('https://stackoverflow.com/')
        request_successful = True
        status_label.config(text="Connection Established", bg="#2ecc71", font="Bahnscrift 10")
    except requests.exceptions.ConnectionError:
        status_label.config(text="No Connection", bg="#e74c3c", font="Bahnscrift 10")
        request_successful = False


Classes

I have never worked much with tkinter, but this problem begs for the use of a class, encapsulating your code with it would make it more modular and would solve the need for a global variable. I will refer you to this tutorial, as it will be more comprehensive than what I know. If you haven't touched object oriented programming in Python yet, I'd recommend reading up on it as well. Below is the basic idea.

import tkinter as tk

import requests


class URLStatus:

    def __init__(self):
        self.session = requests.Session()
        self.request_successful = None
        self.master = tk.Tk()
        ...
        self.status_label = tk.Label(bg="#e67e22", fg="#fff", width=29, height=1, text="Pending", font="Bahnscrift 10")
        self.status_label.place(x=10,y=8)

    def verify_url_status(self):
        try:
            self.session.head('https://stackoverflow.com/')
            self.request_successful = True
            self.status_label.config(text="Connection Established", bg="#2ecc71", font="Bahnscrift 10")
        except requests.exceptions.ConnectionError:
            self.status_label.config(text="No Connection", bg="#e74c3c", font="Bahnscrift 10")
            self.request_successful = False