Here is the unit test solution for testing Nodejs web framework express REST API ONLY USING JEST:
index.js:
const express = require('express');
const { Pool } = require('pg');
const app = express();
const pool = new Pool();
app.post('/insert', (req, res, next) => {
  const values = req.body;
  pool.query(`INSERT INTO student SET ?`, [values], (err, result) => {
    if (err) {
      err = new Error('Not Connected');
      next(err);
    } else {
      res.status(201).json({ msg: `added ${result.insertId}` });
      console.log(result);
    }
  });
});
index.spec.js:
const routes = {};
jest.mock('express', () => {
  const mExpress = {
    post: jest.fn((path, controller) => {
      routes[path] = controller;
    })
  };
  return jest.fn(() => mExpress);
});
let queryCallback;
jest.mock('pg', () => {
  const mpool = {
    query: jest.fn((query, values, callback) => {
      queryCallback = callback;
    })
  };
  const mPool = jest.fn(() => mpool);
  return { Pool: mPool };
});
require('./index');
const express = require('express');
const { Pool } = require('pg');
const app = express();
const pool = new Pool();
describe('insert', () => {
  afterEach(() => {
    jest.restoreAllMocks();
  });
  test('should insert data correctly', done => {
    const logSpy = jest.spyOn(console, 'log');
    expect(app.post).toBeCalledWith('/insert', expect.any(Function));
    const mReq = { body: 1 };
    const mRes = { status: jest.fn().mockReturnThis(), json: jest.fn().mockReturnThis() };
    routes['/insert'](mReq, mRes);
    expect(pool.query).toBeCalledWith('INSERT INTO student SET ?', [1], expect.any(Function));
    const mResult = { insertId: 1 };
    queryCallback(null, mResult);
    expect(mRes.status).toBeCalledWith(201);
    expect(mRes.status().json).toBeCalledWith({ msg: 'added 1' });
    expect(logSpy).toBeCalledWith(mResult);
    done();
  });
  test('should call error handler middleware', () => {
    expect(app.post).toBeCalledWith('/insert', expect.any(Function));
    const mReq = { body: 1 };
    const mRes = { status: jest.fn().mockReturnThis(), json: jest.fn().mockReturnThis() };
    const mNext = jest.fn();
    routes['/insert'](mReq, mRes, mNext);
    expect(pool.query).toBeCalledWith('INSERT INTO student SET ?', [1], expect.any(Function));
    const mError = new Error('network error');
    queryCallback(mError, null);
    expect(mNext).toBeCalledWith(new Error('Not Connected'));
  });
});
Unit test result with 100% coverage:
 PASS  src/stackoverflow/56635460/index.spec.js (7.391s)
  insert
    ✓ should insert data correctly (15ms)
    ✓ should call error handler middleware (1ms)
  console.log node_modules/jest-mock/build/index.js:860
    { insertId: 1 }
----------|----------|----------|----------|----------|-------------------|
File      |  % Stmts | % Branch |  % Funcs |  % Lines | Uncovered Line #s |
----------|----------|----------|----------|----------|-------------------|
All files |      100 |      100 |      100 |      100 |                   |
 index.js |      100 |      100 |      100 |      100 |                   |
----------|----------|----------|----------|----------|-------------------|
Test Suites: 1 passed, 1 total
Tests:       2 passed, 2 total
Snapshots:   0 total
Time:        8.571s
Source code: https://github.com/mrdulin/jest-codelab/tree/master/src/stackoverflow/56635460