I don't like the large number of variables, so I really want to use the exec function.

If the reason is the number of variables
It's definitely better not to use the exec function here.

There is no need to increase the number of variables, manage them in a list or dictionary.
rather,The number of variables is increased by using exec.
It is also not a good use for exec.

If you make it into a function like the code below, you will not be able to get it.

The cause is that tk.BooleanVar is a local variable in the main function.
If you didn't use any functions, they were all global variables.

If you make it a global variable, it will cause malfunction when used elsewhere.
I don't recommend this, but you can see it in action.

exec ('global v {0} \ nv {0} = BooleanVar ()'. format (j + 5 * i + 1))

Addendum: Other changes, root ~ mainloop (), will be moved into the main function.

To make it a local variable, you can specify a dictionary of local variables as an argument to exec.
However, this is also not the recommended solution. The code becomes more complicated.

Use a list or dictionary to manage Checkbuttons and BooleanVar.

How to make multiple checkbuttons

The checkbutton is generated in a loop.

Variables generated including checkbutton are stored in the dictionary.

To identify the checked widget and print its status

In the event before the check status changes, hold down the widget that generated the event.

In the event after the check state changes, the necessary data is fetched from the widget and printed.

from tkinter import *
from tkinter import ttk
root = Tk ()
root.title ('Checkbutton')
frame1 = ttk.Frame (root, padding = (10))
frame1.grid ()
#Before the check status changes
def prev_cb (e):
    # Hold down the widget from which the event occurred
    global cur_cb
    cur_cb = e.widget
#After the check status changes
def after_cb ():
    row, col, var = cbs [cur_cb]
    print (f'{row + 1}, {col + 1} = {var.get ()}')
# Initialize one checkbutton
def init_check (frm, row, col):
    var = BooleanVar ()
    var.set (False)
    cb = ttk.Checkbutton (frm, padding = 10, text = f'{row + 1} x {col + 1}', variable = var, command = after_cb)
    cb.bind ('<1>', prev_cb)
    cb.grid (row = row, column = col)
    return cb, var
cbs = {} #Dictionary for holding widget etc.
cur_cb = None # event checkbutton
for row in range (2): #row
    for col in range (3): #column
        cb, var = init_check (frame1, row, col)
        cbs [cb] = (row, col, var) # Keep necessary variables linked to widget
root.mainloop ()

from tkinter import *
from tkinter import ttk
import functools
root = Tk ()
root.title ('Checkbutton')
frame1 = ttk.Frame (root, padding = (10))
frame1.grid ()
def after_cb (row, col, var):
    print (f'{row + 1}, {col + 1} = {var.get ()}')
# Initialize one checkbutton
def init_check (frm, row, col):
    var = BooleanVar ()
    var.set (False)
    f = functools.partial (after_cb, row, col, var)
    #print (id (f))
    cb = ttk.Checkbutton (frm, padding = 10, text = f'{row + 1} x {col + 1}', variable = var, command = f)
    cb.grid (row = row, column = col)
for row in range (2): #row
    for col in range (3): #column
        init_check (frame1, row, col)
root.mainloop ()