我得到了同样的错误信息（Failed to connect to xxx:1433 - self signed certificate）。然后我使用了trustServerCertificate: true。这为我修复了错误。

我在我的项目中遇到了同样的问题。我设法通过添加trustServerCertificate: true解决了这个问题
示例（NestJS和TypeORM用法）：

TypeOrmModule.forRoot({
  type: 'mssql',
  host: 'localhost\\SQLEXPRESS',
  username: 'sa',
  password: 'root',
  database: 'DB',
  entities: [__dirname + '/**/*.entity{.ts,.js}'],
  synchronize: true,
  extra: {
    trustServerCertificate: true,
  }
}),

对我有效的是，完整的脚本：

var express = require('express');
var app = express();
app.get('/', function (req, res) {
    var sql = require("mssql");
    // config for your database
    var config = {
        user: 'sa',
        password: 'P@ssw0rd',
        server: 'WIN-N4J1057LVFV', 
        database: 'WIN-PAK_ALL',
        synchronize: true,
        trustServerCertificate: true,
    };
    // connect to your database
    sql.connect(config, function (err) {
        if (err) console.log(err);
        // create Request object
        var request = new sql.Request();
        // query to the database and get the records
        request.query('select * from cardHistory', function (err, recordset) {
            if (err) console.log(err)
            // send records as a response
            res.send(recordset);
        });
    });
});
var server = app.listen(5000, function () {
    console.log('Server is running..');
});

如果使用连接字符串：

sql.connect('Server=localhost,1433;Database=database;User Id=username;Password=password;Trusted_Connection=True;TrustServerCertificate=True;');