Here is a bit of a hack I came up with which should do what you want. With the Topology Checker dock widget open and the table populated with some errors, open the Python console, paste the following code into an editor and run. Note* the only thing you need to do is edit the third line of the script to specify the file path to the location where you want the output CSV to be created and what you want it to be named.
Enjoy!
import csv
# Edit line below for your desired output location (the csv will be created)
out_path = 'C:\\Users\\Name\\Desktop\\Topology_errors.csv' # example
check_dock = iface.mainWindow().findChild(QDockWidget, 'checkDock')
main_widget = check_dock.findChild(QWidget, 'dockWidgetContents')
tbl = main_widget.findChild(QTableView, 'mErrorTableView')
out_file = open(out_path, 'w', newline='')
def export_errors(table, output):
writer = csv.writer(output)
mod = table.model()
num_rows = mod.rowCount()
num_cols = mod.columnCount()
headers = []
for n in range(num_cols):
headers.append(mod.headerData(n, Qt.Horizontal))
writer.writerow(headers)
for i in range(num_rows):
errors = []
for j in range(num_cols):
index = mod.index(i, j)
errors.append(mod.data(index))
writer.writerow(errors)
export_errors(tbl, out_file)
out_file.close()