以下はまんま「stackexchange」です。
忘れないようにコピー。
How to perform SQL queries and get results from QGIS python console?

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1) With PyQt4.QtSql: Gary Sherman shows how to query a spatial table in Creating a PostgreSQL Connection From a QGIS Layer Datasource:

from PyQt4.QtSql import *
layer = iface.activeLayer()
uri = QgsDataSourceURI()
uri.setConnection(“localhost”, “5432”, “testpostgis”, “me”, “”)
uri.setDataSource(“public”, “teststrati”, “the_geom”)
# add the layer to the canvas
vlayer = QgsVectorLayer(uri.uri(), “tot”, “postgres”)
# now query the table
db = QSqlDatabase.addDatabase(“QPSQL”);
db.setDatabaseName(uri.database())
db.setPort(int(uri.port()))
db.setUserName(uri.username())
db.setPassword(uri.password())
db.open()
# query the table
query = db.exec_(“””select * from teststrati”””)
query.next()
query.value(0)
130
# etc read the documentation of QtSQL
And you can open all the other tables/views (spatial or non spatial) similarly:

db = QSqlDatabase.addDatabase(“QPSQL”)
db.setHostName(“localhost”)
db.setPort(5432)
# non spatial table or view
db.setDatabaseName(“people”)
db.setUserName(“me”)
db.setPassword(“”)
query = QSqlQuery(db)
query.exec_(“select * from people;”)
# etc.
2) Or you can use the standard Python module for PostgreSQL/PostGIS: Psycopg2:

import psycopg2
conn = psycopg2.connect(“dbname=’testpostgis’host=’localhost’ user=’me'”)
cur = conn.cursor()
sql = “””SELECT “DIP_DIR”,”DIP”, ST_AsGeoJSON(the_geom) from teststrati;”””
cur.execute(sql)
result = cur.fetchone()
print result
(130, 30, u'{“type”:”Point”,”coordinates”:[272070.600040999997873,155389.387920000008307]}’)
With a non spatial table or view and the result as a dictionary:

conn = psycopg2.connect(“dbname=’testpostgis’host=’localhost’ user=’me'”)
dict_cur = conn.cursor(cursor_factory=psycopg2.extras.DictCursor)
dict_cur.execute(“””SELECT * from people;”””)
rec = dict_cur.fetchone()
rec.keys()
[‘name’, ‘id’]
rec.values()
(‘Jon Doe’,1)
# etc read the documentation of the module

class HelloWorld:
    def __init__(self, iface):
        # Save reference to the QGIS interface
        self.iface = iface
        self.canvas = iface.mapCanvas()
        QgsMessageLog.logMessage("__init__", 'HelloWorld', QgsMessageLog.INFO)

QGISのログパネルに出力される

QGIS3ではこんなかんじ

from qgis.core import Qgis, QgsMessageLog
import configparser
import os

#======================================================================
# ログクラス
#======================================================================
class Trace:
    def __init__(self):
        print('init')
    @staticmethod
    def info(tag, msg):
        QgsMessageLog.logMessage(msg, tag, Qgis.Info)

QGISのソースを確認すると「QgsDebugMsg」で多くのログが吐かれている。
これはQGISの内部の動作を追うには大変参考になる。
※レンダリング関係はデバッカーで追うのは大変。

実際にqgslogger.cppを確認すると下記の環境変数を設定する必要がある。
QGIS_DEBUG：1
QGIS_LOG_FILE：ログファイルのフルパス
※QGIS_DEBUG_FILEは不明

int QgsLogger::sDebugLevel = -999; // undefined value
int QgsLogger::sPrefixLength = -1;
QString QgsLogger::sFileFilter;
QString QgsLogger::sLogFile;
QTime QgsLogger::sTime;

void QgsLogger::init()
{
  if ( sDebugLevel != -999 )
    return;

  sTime.start();

  sLogFile = getenv( "QGIS_LOG_FILE" ) ? getenv( "QGIS_LOG_FILE" ) : "";
  sFileFilter = getenv( "QGIS_DEBUG_FILE" ) ? getenv( "QGIS_DEBUG_FILE" ) : "";
  sDebugLevel = getenv( "QGIS_DEBUG" ) ? atoi( getenv( "QGIS_DEBUG" ) ) :
#ifdef QGISDEBUG
                1
#else
                0
#endif
                ;

  sPrefixLength = sizeof( CMAKE_SOURCE_DIR );
  if ( CMAKE_SOURCE_DIR[sPrefixLength-1] == '/' )
    sPrefixLength++;
}

void QgsLogger::logMessageToFile( const QString& theMessage )
{
  if ( sLogFile.isEmpty() )
    return;

  //Maybe more efficient to keep the file open for the life of qgis...
  QFile file( sLogFile );
  if ( !file.open( QIODevice::Append ) )
    return;
  file.write( theMessage.toLocal8Bit().constData() );
  file.write( "\n" );
  file.close();
}

qgis起動バッチ(qgis.bat）に以下を記述してQGISを起動すると指定されたログが出力される

set QGIS_LOG_FILE=C:\QGIS\log.txt
set QGIS_DEBUG=1