"In astonishment mingled with terror the king listened to the message from the man whom he feared and hated, and for whom he had sought so untiringly. Well he knew that Elijah would not endanger his life merely for the sake of meeting him. Could it be possible that the prophet was about to utter another woe against Israel? The king's heart was seized with dread. He remembered the withered arm of Jeroboam. Ahab could not avoid obeying the summons, neither dared he lift up his hand against the messenger of God. And so, accompanied by a bodyguard of soldiers, the trembling monarch went to meet the prophet. PK 139.2"
"As the Lord thy God liveth, there is no nation or kingdom, whither my lord hath not sent to seek thee: and when they said, He is not there; he took an oath of the kingdom and nation, that they found thee not."
Both passages deal with Ahab's fear of Elijah. 1 Kings 18:10 shows how extensively Ahab sought Elijah. The EGW passage describes Ahab's fear and dread upon hearing that Elijah wanted to meet him. The connection lies in Ahab's attitude towards Elijah and the context of their encounter.
"And it came to pass, when king Jeroboam heard the saying of the man of God, which had cried against the altar in Beth–el, that he put forth his hand from the altar, saying, Lay hold on him. And his hand, which he put forth against him, dried up, so that he could not pull it in again to him."
Both passages involve a king (Ahab in EGW, Jeroboam in 1 Kings) being reminded of a previous divine judgment. The EGW passage explicitly mentions Ahab remembering the withered arm of Jeroboam, while 1 Kings 13:4 describes the event itself. This memory influences Ahab's reaction to Elijah. The connection is the shared element of a king being confronted with divine power and remembering a past judgment.
"The king and the prophet stand face to face. Though Ahab is filled with passionate hatred, yet in the presence of Elijah he seems unmanned, powerless. In his first faltering words, “Art thou he that troubleth Israel?” he unconsciously reveals the inmost feelings of his heart. Ahab knew that it was by the word of God that the heavens had become as brass, yet he sought to cast upon the prophet the blame for the heavy judgments resting on the land. PK 139.3"
"And it came to pass, when Ahab saw Elijah, that Ahab said unto him, Art thou he that troubleth Israel?"
The EGW passage directly quotes and comments on 1 Kings 18:17, making it a clear explicit allusion.
"It is natural for the wrongdoer to hold the messengers of God responsible for the calamities that come as the sure result of a departure from the way of righteousness. Those who place themselves in Satan's power are unable to see things as God sees them. When the mirror of truth is held up before them, they become indignant at the thought of receiving reproof. Blinded by sin, they refuse to repent; they feel that God's servants have turned against them and are worthy of severest censure. PK 139.4"
"And this is the condemnation, that light is come into the world, and men loved darkness rather than light, because their deeds were evil."
Both passages discuss how people react negatively to truth and prefer to remain in darkness/sin. The EGW passage expands on the idea of blaming the messenger, which is a consequence of loving darkness rather than light.