Traditionally, Calvinist / Reformed Christians have not viewed the process of regeneration as God "forcing" a person to become a Christian, since although God alone is the initiator of the process, the person being regenerated cooperates with it, and also because the word "force" has a misleading negative connotation given that salvation is a restorative and liberating process, while "force" typically implies the application of constraints.
This is a mystery that can't be fully explained but relates to the overall mystery of how divine sovereignty and human responsibility relate.
Note: I noticed you appear to be asking two related questions. I've primarily answered the word about whether Calvinist would say people are "forced to be saved against their will", and why Calvinists would deny that.
I've not discussed as much your question of whether effectual calling forces people to be saved some because they naturally were opposed to God prior to receiving the call. In brief, I would say that God does go against the will of the unregenerate in that sense, but that the unregenerate were created in order to find their delight in God (see Westminster Shorter Catechism Q1) so this is a deliverance for them from bondage to sin and so it is not a cause for criticism that God elects to go against their wills, since otherwise none could be saved (cf. Rom. 2-3)
Below I mainly discuss the mode of effectual calling, and why calling it God forcing us to be saved is misleading despite His sovereignty.
Regarding human will in general, Calvinists do believe people have "natural liberty" of will (Matt. 17:12, Jas. 1:14, Dt. 30:19) so that they are not "forced" to do good or evil. (Westminster Confession of Faith 9.1). This is not viewed as conflicting with God's eternal decree of "whatsoever comes to pass (Eph. 1:11, Rom. 9:15,18)": despite being sovereign, God is "not the author of sin (Jas. 1:13), nor is violence offered to the will of the creature (Matt. 17:12)" [all Scripture references from Confession].
Calvinists are commonly known as not believing in free will since historically speaking in the Reformation era, free will meant the soul had the ability to seek God and profess saving faith without the prior act of God in regeneration. (See Luther's work on free will for example.) Calvinists do not believe people who are not Christian have free will in that sense, since they are "dead in transgressions and sins" until Christ makes them alive (Eph. 2:1, 4-6, 8-10).
However, the Westminster Confession of Faith states that the "effectual calling" (the Spirit's inner call on the soul the makes the outer Gospel call salvific), which "renew[s] the will and by [God's] almighty power, determine[es it] to what is good" is one whereby people come most freely, being made willing by His grace."
The Scripture references that the Westminster Confession uses for the people coming freely are Song of Solomon 1:4 (with the King understood as being Christ), Ps. 110:3, John 6:37, Rom. 6:16-16.
Of these, Ps. 110:3 is cited most often by Reformed writers in this context. In Hebrew, it can be translated either as "your people shall be made willing in the day of your power" or "your people shall offer themselves freely in the day of your power": the mystery of salvation is that both are true. Ps. 110 is used both by Jesus in the Synoptic Gospels (Matthew, Mark, Luke) and by the author of Hebrews as a Messianic Psalm (a Psalm about Jesus). In the context of 110:3 it refers to the beauty of the King/Messiah. Traditionally, Reformed writers have taken this to mean that God reveals Himself inwardly in conversion in His moral beauty as infinitely worthy, and that this spiritual sight of Christ is of the essence of effectual calling.
Though in Reformed theology, people are "totally depraved", in the sense that their wills apart from God will always tend toward evil (self-centeredness and doing good for the wrong motives being forms of evil), they do believe that people are still made in the image of God, and thus we have an innate desire for something good, even though we pervert this desire into worship of the creature not the Creator (cf. Rom. 1). As St. Augustine, the great defender of monergism (God's sovereignty in salvation) said, "Our hearts are restless until they find their rest in You."
In effectual calling, God shows Himself, not your own self, not the creation, to be the only true source of rest, and the person who is being saved acknowledges the truth, since the "eyes of the heart have been opened" (Eph. 1:18). Their bondage to sin is taken away so they can freely serve Christ. The language of "forcing" is misleading here since people apart from Christ are the ones in bondage to sin, and God is restoring their wills in salvation to how they were originally created to be (though not perfectly until the Resurrection, WCF 10.5)
18th-century Reformed theologian Jonathan Edwards discusses at great length in his Religious Affections how salvation is based on a spiritual sight of the moral beauty and excellency of Christ. John Piper, who is highly influenced by Edwards, discusses this using the Eph. 1:18 and Room. 1 passages referenced above here: https://www.desiringgod.org/articles/how-god-opens-the-eyes-of-the-heart.
Finally, John Calvin himself said regarding John 6:44, "God's drawing is not violent, so as to compel men by external force" (Commentary on the Gospels). Puritan theologian John Gill said God's drawing is an "act of power but not of force".
The distinction is important here since an act of force would suggest a violation of the human person, whereas salvation as described in the Bible is actually redemption of them from slavery to sin, restoration of the image of God (a new creation), and enlightenment to the truth.
As it says in 1 John, "We love because He first loved us." God's love is the precondition for ours, but, because it is love, coercive words like "force" are misleading at best.
It is true that some Reformed people have even used phrases like "holy rape" to describe salvation. (Or more poetically expressed, the words of John Donne's "Batter my heart Three-Personed God".) However, I believe that this is due to their trying to resolve the mystery of how divine sovereignty and human responsibility relate, rather than being faithful to the balance of the Scripture's teaching.
I believe my citations from the Westminster Confession, Gill, Calvin, and Piper show that most people in the Reformed tradition would not use the word "force" to describe effectual calling / regeneration, due to it's connotations. God is in control, but the person being saved is cooperating as God's grace acts upon him or her.